Fourier's Law for a Harmonic Crystal with Self-consistent Stochastic Reservoirs

TL;DR

This paper rigorously proves Fourier's law for a d-dimensional harmonic crystal with self-consistent stochastic reservoirs, showing the existence of a unique steady state and computing the thermal conductivity, which aligns with classical results in the literature.

Contribution

It establishes the existence and uniqueness of the steady state for the self-consistent harmonic crystal model and computes the thermal conductivity using Green-Kubo formula.

Findings

The steady state is unique and exhibits local thermal equilibrium.

The heat current satisfies Fourier's law with a positive finite conductivity.

The conductivity matches classical results for the 1D case and scales as 1/(l_d*d) for large dimensions.

Abstract

We consider a d-dimensional harmonic crystal in contact with a stochastic Langevin type heat bath at each site. The temperatures of the "exterior" left and right heat baths are at specified values T_L and T_R, respectively, while the temperatures of the "interior" baths are chosen self-consistently so that there is no average flux of energy between them and the system in the steady state. We prove that this requirement uniquely fixes the temperatures and the self consistent system has a unique steady state. For the infinite system this state is one of local thermal equilibrium. The corresponding heat current satisfies Fourier's law with a finite positive thermal conductivity which can also be computed using the Green-Kubo formula. For the harmonic chain (d=1) the conductivity agrees with the expression obtained by Bolsterli, Rich and Visscher in 1970 who first studied this model. In the other limit, d>>1, the stationary infinite volume heat conductivity behaves as 1/(l_d*d) where l_d is the coupling to the intermediate reservoirs. We also analyze the effect of having a non-uniform distribution of the heat bath couplings. These results are proven rigorously by controlling the behavior of the correlations in the thermodynamic limit.